When people say “anti-woke”, they actually mean that they are anti-doing anything about the awareness of systemic inequality that wokeness indicates. By definition, someone who is against change/progress is a conservative, so when someone says they are anti-woke, they are by definition expressing a conservative stance. That is, wanting to do something about systemic inequality is synonymous with having a progressive stance on systemic inequality.
Being a tankie, on the other hand, is not synonymous with being a comunist. Tankies are just one form of communist (militant).